Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

A mean-field theory approach to the Ising-model gives a critical temperature $k_B T_C = q J$, where $q$ is the number of nearest neighbours and $J$ is the interaction in the Ising Hamiltonian. Setting $q = 2$ for the 1D case gives $k_B T_C = 2 J$. Based on this argument there would be a phase transition in the 1D Ising model. This is obviously wrong.

Is mean-field-theory invalid for the 1D case? Am I missing something here?

share|improve this question

1 Answer 1

up vote 5 down vote accepted

Yes mean-field theory is wrong for the one-dimensional case (and wrong for the two and three dimensional cases as well, where the transition exists but the mean-field approximation gets the wrong critical temperature and exponents). In fact it's a typical first year exercise to solve the 1D Ising model exactly using transfer matrices, and I suggest you look into that.

The nature of the mean-field approximation is that it assumes there are no thermal fluctuations around the approximate solution you propose (i.e., a state with ferromagnetic order) but in low dimensions, this approximation is often qualitatively wrong.

The mean-field theory of the Ising model happens to be exact in 4-dimensions, but more complicated phase transitions might not be well described by mean-field theory for even higher dimensions (this is called the "upper critical dimension").

share|improve this answer
    
Is there an intuitive explanation as to why the thermal fluctuations play a more important role with fewer dimensions? –  lucas clemente Jul 23 '12 at 14:12

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.